%0 Journal Article
%T Some Existence Theorems for Nonconvex Variational Inequalities Problems
%A Narin Petrot
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/472760
%X By using nonsmooth analysis knowledge, we provide the conditions for existence solutions of the variational inequalities problems in nonconvex setting. We also show that the strongly monotonic assumption of the mapping may not need for the existence of solutions. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature. 1. Introduction Variational inequalities theory, which was introduced by Stampacchia [1], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems. It should be pointed out that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting; see [2¨C5] for examples. Moreover, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping. In fact, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined, even in the nonconvex case (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.). The main aim of this paper is intending to consider the conditions for the existence solutions of some variational inequalities problems in nonconvex setting. We will make use of some recent nonsmooth analysis techniques to overcome the difficulties that arise from the nonconvexity. Also, it is worth mentioning that we have considered when the mapping may not satisfy the strongly monotonic assumption. In this sense, our result represents an improvement and refinement of the known results. 2. Preliminaries Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed subset
%U http://www.hindawi.com/journals/aaa/2010/472760/